A Concordance Invariant For Knots from $CF{K}^{\mathrm{\infty }}$

We construct a multi-filtered smooth knot concordance invariant, $\beta (t^1,a,t^2)$, derived from the knot Floer complex $CF{K}^{\mathrm{\infty }}$. $\beta$ is a three-variable piecewise linear function defined on $(0,2)\times \mathbb{R}\times (0,2)$. We demonstrate that $\beta$ satisfies a subadditivity relation. We apply $\beta$ to prove concordance results for a certain family of L-space knots. We show that the positive cones spanned by Teragaito’s knots $K_{1,n}$ and $K_{2,n}$ are disjoint in the smooth concordance group $\mathcal{C}$. For positive L-space knots, we show $\beta$ completely determines the Alexander polynomial and distinguishes knots with identical $\mathrm{{\rm Y}}$ invariants.